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# sols2 - 2 SEPTEMBER 1ST LOGIC 3 2 September 1st Logic 2.1...

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2. SEPTEMBER 1ST: LOGIC 3 2. September 1st: Logic 2.1 . Prove De Morgan’s laws. Answer: Recall that these say ¬ ( p q ) ⇐⇒ ( ¬ p ) ( ¬ q ) ¬ ( p q ) ⇐⇒ ( ¬ p ) ( ¬ q ) . We can take care of these with two truth tables: p q ¬ p ¬ q p q ¬ ( p q ) ( ¬ p ) ( ¬ q ) T T F F T F F T F F T T F F F T T F T F F F F T T F T T proves the first law, and p q ¬ p ¬ q p q ¬ ( p q ) ( ¬ p ) ( ¬ q ) T T F F T F F T F F T F T T F T T F F T T F F T T F T T proves the second. Note that the second law actually follows from the first (or conversely, if you prefer), so one table would su ffi ce. To see this, assume you have proved the first law; negate both sides to get p q ⇐⇒ ¬ (( ¬ p ) ( ¬ q )) ; then apply this tautology to p = ¬ r and q = ¬ s : ( ¬ r ) ( ¬ s ) ⇐⇒ ¬ ( r s ) . Of course r and s are just names for arbitrary statements, so the last tautology we obtained is just a restatement of the second law. 2.2 . Assume that L ( x ) means x is a lion’; C ( x ) means x is a cow’; R ( x ) means x is red’; G ( x

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sols2 - 2 SEPTEMBER 1ST LOGIC 3 2 September 1st Logic 2.1...

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